Curl and gradient of divergence operators in Spaces ${\mathbf{W}}^m$ and ${\mathbf{A}}^{2k}$ vortex and potential fields and in the classes $\mathbf{C}(2k,m)$

Abstract

The properties of the curl and the gradient of divergence operators ( $\text{rot}$ and $\nabla \text{div}$ ) are studied in the space ${\mathbf{L}}_2(G)$ in a bounded domain $G\subset {\text{R}}^3$ with a smooth boundary $\Gamma$ and in the classes $\mathbf{C}(2k,m)(G)\equiv {\mathbf{A}}^{2k}(G)\oplus {\mathbf{W}}^m(G)$. The space ${\mathbf{L}}_2(G)$ is decomposed into orthogonal subspaces $\mathcal{A}$ and $\mathcal{B}$: ${\mathbf{L}}_2(G)=\mathcal{A}\oplus \mathcal{B}$. In turn, $\mathcal{A}={\mathcal{A}}_H\oplus {\mathbf{A}}^0$ and $\mathcal{B}={\mathcal{B}}_H\oplus {\mathbf{V}}^0$, where ${\mathcal{A}}_H$ and ${\mathcal{B}}_H$ are null spaces of operators $\nabla \text{div}$ and $\text{rot}$ in $\mathcal{A}$ and $\mathcal{B}$; the dimensions of ${\mathcal{A}}_H$ and ${\mathcal{B}}_H$ are finite and determined by the topology of the boundary; ${\mathcal{A}}_H=\varnothing$ and ${\mathcal{B}}_H=\varnothing$ if the domain $\Omega$ is a ball. The orthonormal basis are constructed in the class ${\mathbf{A}}^0$ (resp., In ${\mathbf{V}}^0$ ) by eigenfields ${\mathbf{q}}_j(\mathbf{x})$ of $\nabla \text{div}$ operator (resp., ${\mathbf{q}}_j^{\pm }(\mathbf{x})$ of $\text{rot}$ operator) with nonzero eigenvalues ${\mu }_j$ (resp., $\pm {\lambda }_j$ ). The operators $\nabla \mathrm{div}$ and $\mathrm{rot}$ cancel each other out and project ${\mathbf{L}}_2(G)$ onto $\mathcal{A}$ and $\mathcal{B}$, and $\mathrm{rot}\mathbf{u}=0$ for $\mathbf{u}\in \mathcal{A}$, and $\nabla \mathrm{d}iv\mathbf{v}=0$ for $\mathbf{v}\in \mathcal{B}$ \cite{hw}. Laplace matrix operator expressed through them: $\mathrm{\Delta }\mathbf{v}\equiv \nabla \mathrm{div}\mathbf{v}-(\mathrm{rot}{)}^2\mathbf{v}$.

26 pages, in Russian